Probabilistic Zeta Functions of Finite Groups Under Coprime Actions

Let a finite group [Formula: see text] act on a finite group [Formula: see text] via automorphism and let [Formula: see text] be a positive integer. In this paper, we introduce the probabilistic zeta function of ordered [Formula: see text]-tuples from [Formula: see text] which [Formula: see text]-generate [Formula: see text], denoted by [Formula: see text]. When [Formula: see text], it is proved that [Formula: see text] is solvable if and only if [Formula: see text] has an Euler product expansion with all factors of the form [Formula: see text], where each [Formula: see text]is a prime power. This is a generalization of Detomi and Lucchini's result in 2003.